©2025 Ada Academica https://adac.eeEur. J. Math. Anal. 5 (2025) 2doi: 10.28924/ada/ma.5.2
On η-Local Functions in Ideal Topological Spaces

Junvon A. Almocera∗, Lezel M. Tutanes
Department of Mathematics, College of Arts and Sciences, Bukidnon State University, Malaybalay City,

Bukidnon, Philippines
1901103646@student.buksu.edu.ph, lezeltutanes@buksu.edu.ph

∗Correspondence: 1901103646@student.buksu.edu.ph

Abstract. This study introduces and investigates a new local function called η-local function in idealtopological space (X, τ, I) by using the notion of η-open sets in topological space (X, τ). Theoperator (·)∗η : P(X) → P(X) is defined as (·)∗η(A) = A∗η =
{
x ∈ X : A ∩ U /∈ I for every U ∈

η-O(x)
} for each A ⊆ X , where η-O(x) is the set of all η-open subset of X containing x . Thisstudy establishes some properties of A∗η including its relationships to the local function and localfunction Γ∗ in ideal topological space (X, τ, I). This study also introduces a new type of closurecalled the η-local closure in ideal topological space (X, τ, I) which is denoted by Cl∗η(A) for each

A ⊆ X . Furthermore, this study establishes some properties of the η-local closure.

1. Introduction

The concept of ideal topological spaces was first studied by Kuratowski [4] and Vaidyanathas-wamy [9]. The notion of topological spaces with ideals were investigated by Jankovic [3]. Thereafter,the study of ideal topological spaces attracts the attention of many topologists. Recently, Al-omari [1] have introduced and investigated the notion of local function Γ∗ in an ideal topologicalspace and showed that Γ∗ is equivalent to the δ-local function due to Hatir et al. [2]. In this paper,the researcher defined a new type of local function called the η-local function in ideal topologicalspaces by using the η-open set of Subbulakshmi [8] and established some of its properties, includingits relationship to the local function and local function Γ∗ in ideal topological spaces. Subsequently,the η-local closure has been defined, and some of the properties are established.
2. Preliminaries

Throughout this paper (X, τ) and (X, τ, I) denote a topological space and an ideal topologicalspace, respectively. The members of τ are called open sets and their complement are calledclosed sets. For any subset A of X , the closure and interior of A are denoted by cl(A) and
Received: 4 Jun 2024.
Key words and phrases. Ideal topoloical space, η-open set; η-local function; η-local closure.1

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 2

int(A), respectively. A subset A of a space (X, τ) is said to be η-open (resp. η-closed) [8] if
A ⊆ int(cl(int(A))) ∪ cl(int(A)) (resp. A ⊇ cl(int(cl(A))) ∩ int(cl(A))). The η-closure of A isdefined by the intersection of all η-closed sets containing the set A and it is denoted by η-cl(A) [8].A subset A of a space (X, τ) is said to be semi-open [6] if A ⊆ cl(int(A)). A subset A of a space
(X, τ) is said to be regular-open [7] if A = int(cl(A)). The familiy of all η-open (resp. semi-open,regular open) sets in X is denoted by η-O(X) (resp. SO(X), RO(X)).An ideal I [5] on a topological spaces (X, τ) is a nonempty collection of subsets of X , whichsatisfies (i) A ∈ I and B ∈ I implies A ∪ B ∈ I; and (ii) A ∈ I and B ⊆ A implies B ∈ I . Then thetriplet (X, τ, I) is called an ideal topological space. If P(X) is the set of all subsets of X , a setoperator (·)∗ : P(X)→ P(X) called a local function [3,9] of A with respect to τ and I is defined asfollows: for A ⊆ X , A∗(I, τ) = {x ∈ X : U∩A /∈ I, for every U ∈ τ(x)} where τ(x) = {U ∈ τ : x ∈
U}. A∗(I, τ) can simply be written as A∗. For every ideal topological space, there exists a topology
τ∗(I, τ) or briefly τ∗ [3], finer than τ , generated by the B(I, τ) = {U \ J : U ∈ τ and J ∈ I},however, B(I, τ) is not a topology in general. Additionally, Cl∗(A) = A∪A∗ defines a Kuratowskiclosure operator [5] for τ∗. A subset A of an ideal topological spaces is τ∗-closed set or ∗-closedset [3] if A∗ ⊆ A. Let (X, τ, I) be an ideal topological spaces and A be a subset of X . Then
Γ∗(A)(I, τ) = {x ∈ X : A∩U /∈ I, for every U ∈ RO(X)} where RO(X) = {U ∈ RO(X) : x ∈ U}.
Γ∗(A)(I, τ) can simply be denoted as Γ∗(A) [1].

3. η-local functions

Definition 1. Let (X, τ, I) be an ideal topological space. Then the operator (·)∗η : P(X)→ P(X) is
defined as for A ⊆ X , A∗η

(
I, η-O(X)

)
= {x ∈ X : A ∩ U /∈ I, for every U ∈ η-O(x)} where η-O(x) =

{U ∈ η-O(X) : x ∈ U} is called the η-local f unction of A with respect to I and η-O(X).
A∗η
(
I, η-O(X)

)
can simply be denoted by A∗η .

Example 1. Let (X, τ, I) be an ideal topological where X = {a, b, c, d}, τ = {X,∅, {a}, {b}, {a, b},
{a, b, c}, {a, b, d}}, and I = {∅, {c}, {d}, {c, d}}. Then η-O(X) = {∅, X, {a}, {b}, {a, b}, {a, c},
{a, d}, {b, c}, {b, d}, {a, b, c}, {a, c, d}, {a, b, d}, {b, c, d}}. Now, let A = {a, b, d}. Then by
Definition 1, A∗η = {a, b, c, d} = X .

Theorem 1. Let (X, τ, I) be an ideal topological space and A,B be subsets of X . Then for any
η-local functions, the following properties hold:

(i) if A ⊆ B, then A∗η ⊆ B∗η;
(ii) (A ∩ B)∗η ⊆ A∗η ∩ B∗η; and

(iii) A∗η ∪ B∗η ⊆ (A ∪ B)∗η;
(iv) if A = ∅, then A∗η = ∅;
(v) if A∗η ∩ B /∈ I , then A∗η ∩ B 6= ∅;

(vi)
(
A∗η
)∗
η
⊆ A∗η;

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 3

(vii) if A ∈ I , then A∗η = ∅;
(iix) if I = {∅}, then A∗η = η-cl(A);
(ix) if I = P(X), then A∗η = ∅; and
(x) A∗η ⊆ η-cl(A);

Proof.

(i) Let A,B ⊆ X and A ⊆ B. Suppose x /∈ B∗η , then there exist U ∈ η-O(x) such that
B ∩ U ∈ I . Since A ⊆ B, A ∩ U ⊆ B ∩ U ∈ I , by Definition of ideal, A ∩ U ∈ I . Hence,
x /∈ A∗η .

(ii) Let A,B ⊆ X . Since A ∩ B ⊆ A and A ∩ B ⊆ B, by Theorem 1 (i), (A ∩ B)∗η ⊆ A∗η and
(A ∩ B)∗η ⊆ B∗η , respectively. Hence, (A ∩ B)∗η ⊆ A∗η ∩ B∗η .

(iii) Let A,B ⊆ X . Since A ⊆ A ∪ B and B ⊆ A ∪ B, by Theorem 1 (i), A∗η ⊆ (A ∪ B)∗η and
B∗η ⊆ (A ∪ B)∗η , respectively. Hence, A∗η ∪ B∗η ⊆ (A ∪ B)∗η .

(iv) Let A = ∅. Suppose A∗η 6= ∅. Then there exists x ∈ A∗η . It follows that A ∩ U = ∅ ∩ U =

∅ /∈ I for every U ∈ η-O(x). Since I is an ideal, ∅ ∈ I which is a contradiction.
(v) Let A∗η ∩ B /∈ I . Suppose A∗η ∩ B = ∅. Note that by definition of ideal, ∅ ∈ I for anyideal I . Now, since A∗η ∩ B = ∅ and I is an ideal, A∗η ∩ B = ∅ ∈ I implies A∗η ∩ B ∈ I , acontradiction.

(vi) Let x ∈ (A∗η)∗η . Then, for every U ∈ η-O(x), U ∩ A∗η /∈ I and hence, by (ii), U ∩ A∗η 6= ∅.Now, let y ∈ U ∩A∗η . Then, U ∈ η-O(y) and y ∈ A∗η . Hence, we have U ∩A /∈ I . Note that
U ∈ η-O(x) and U ∩ A /∈ I . It follows that x ∈ A∗η . Therefore, (A∗η)∗η ⊆ A∗η .

(vii) Let A ∈ I . Suppose A∗η 6= ∅. Then there exists an element x ∈ A∗η . Then A ∩ U /∈ I forevery U ∈ η-O(x). Now, Since, A ∩ U ⊆ A ∈ I and I is an ideal, A ∩ U ∈ I which is acontradiction.
(iix) Let I = {∅}. Suppose that A∗η 6= η-cl(A). Let η-cl(A) ⊂ A∗η , then there exists an element

x ∈ A∗η and x /∈ η-cl(A). It follows that for every A ⊆ X , since x ∈ A∗η , A∩U /∈ I for every
U ∈ η-O(x). Since I = {∅}, A∩U 6= ∅ for every U ∈ η-O(x). Note that U ∈ η-O(x) means
x ∈ U where U is η-open set. Since, x /∈ η-cl(A), x /∈ ⋂{K : K is η-closed and A ⊆ K}.It follows that x /∈ K for some η-closed set K such that A ⊆ K. Hence, x ∈ Kc for some
η-open set Kc such that A ∩Kc = ∅. It implies that there exists an η-open set Kc where
x ∈ Kc and A ∩Kc = ∅, a contradiction.

(ix) Let I = P(X). Note that A ⊆ X , then A ∈ P(X). Since I = P(X), A ∈ I . Hence, byTheorem 1 (vii), A∗η = ∅.
(x) Let x /∈ η-cl(A). Then, x /∈ ⋂{K : K is η-closed and A ⊆ K}. It follows that x /∈ K forsome η-closed set K such that A ⊆ K. Hence, x ∈ Kc for some η-open set Kc such that

A ∩ Kc = ∅. It implies that there exists Kc ∈ η-O(x) such that A ∩ Kc = ∅, and bydefinition of ideal, ∅ ∈ I for any ideal I . Hence, A ∩ Kc ∈ I for some Kc ∈ η-O(x). Thisshows that x /∈ A∗η . �

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 4

Remark 1. Let (X, τ, I) be an ideal topological space and A be any subset of X . Then for any
η-local functions, the following properties hold:(i) The reverse inclusion of Theorem 1 (iii) need not be true in general.(ii) Neither A ⊆ A∗η nor A∗η ⊆ A in general.(iii) A∗η is an η-closed set iff A∗η = η-cl(A∗η).

In order to verify Remark 1 (i) and (ii), the following examples are shown.
Example 2.(i) Consider the ideal topological space (X, τ, I), where X = {a, b, c, d}, τ = {X,∅, {b}, {c},

{b, c}, {a, b, c}}, and I = {∅, {a}}. Then the η-open sets are ∅, X , {b}, {c}, {a, b},
{a, c}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, c, d}, {a, b, d}, and {b, c, d}. Now, let A = {b}
and B = {c}, then A ∪ B = {b, c}. Then by applying Definition 1, A∗η = {b}, B∗η = {c},
and (A ∪ B)∗η = X . Observe that (A ∪ B)∗η = X and A∗η ∪ B∗η = {b, c}. These shows that
(A ∪ B)∗η * A∗η ∪ B∗η .(ii) Consider the ideal topological space (X, τ, I), where X = {a, b, c, d}, τ = {X,∅, {b}, {c},
{b, c, d}}, and I = {∅, {c}}. Then the η-open sets are ∅, X , {b}, {c}, {a, b}, {b, c},
{b, d}, {a, b, c}, {a, b, d}, and {b, c, d}. Let A,B ⊂ X where, A = {a, c, d} and B = {a, b}.
Then by Definition 1, A∗η = {a, d} and B∗η = {a, b, d}. Obeserve that A * A∗η and B∗η * B.

Theorem 2. Let (X, τ, I) be an ideal topological space and A,B be subsets of X . Then for any
η-local functions, the following properties hold:

(i) (A \ B)∗η \ B∗η ⊆ A∗η \ B∗η;
(ii) if B ∈ I , then (A ∪ B)∗η = A∗η = (A \ B)∗η;

(iii) (A \ B)∗η ∪ (B \ A)∗η ⊆ (A ∪ B)∗η;
(iv) if U ⊆ X , then U ∩ (U ∩ A)∗η ⊆ U ∩ A∗η;
(v) if U ∈ I , then (A ∩ U)∗η = ∅;

(vi) if A is an η-closed set, then A∗η ⊆ A;
(vii)

(
A ∩ A∗η

)∗
η
⊆ A∗η;

(iix) if A ∪ B ∈ I , then (A ∪ B)∗η = A∗η ∪ B∗η = ∅.
(ix) A∗η = η-cl(A∗η) ⊆ η-cl(A) and A∗η is an η-closed set; and
(x) if A ⊆ A∗η , then A∗η = η-cl(A∗η) = η-cl(A).

Proof.

(i) Let A,B ⊆ X . Since A\B ⊆ A, by Theorem 1 (i), (A\B)∗η ⊆ A∗η implies that (A\B)∗η\B∗η ⊆
A∗η \ B∗η .

(ii) Let B ∈ I . Suppose x ∈ (A∪B)∗η . Then for every U ∈ η-O(x), (A∪B)∩ U /∈ I . Note that
(A∩U)∪(B∩U) = (A∪B)∩U /∈ I implies that (A∩U)∪(B∩U) /∈ I . It shows that A∩U /∈ I

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 5or B∩U /∈ I , or both, and as a result, x ∈ A∗η or x ∈ B∗η , or both. Hence, x ∈ A∗η∪B∗η . Now,note that B ∈ I , then by Theorem 1 (vii), B∗η = ∅. Thus, x ∈ A∗η ∪B∗η = A∗η ∪∅ = A∗η . Thisimplies that x ∈ A∗η . Hence, it shows that (A∪B)∗η ⊆ A∗η . In contrast, since A ⊆ A∪B, byTheorem 1 (i), it implies that A∗η ⊆ (A ∪ B)∗η . Consequently, as a result, A∗η = (A ∪ B)∗η .Now, suppose that (A\B)∗η 6= A∗η . Let (A\B)∗η ⊂ A∗η . Then there exists an element x ∈ A∗ηsuch that x /∈ (A \B)∗η . Note that x ∈ A∗η implies that for every A ⊆ X , A∩U /∈ I for every
U ∈ η-O(x). Now, since x /∈ (A \B)∗η , there exists U ∈ η-O(x) such that (A \B) ∩ U ∈ I .Note that (A \ B) ∩ U = (A ∩ U) \ (B ∩ U) ∈ I and B ∩ U ⊆ B ∈ I . Now, since I is anideal, B ∩ U ∈ I and [(A ∩ U) \ (B ∩ U)

]
∪ (B ∩ U) ∈ I , respectively. Again, note that[

(A∩U)\(B∩U)
]
∪(B∩U) = (A∩U)∪(B∩U) ∈ I . Since (A∩U) ⊆ (A∩U)∪(B∩U) ∈ I ,it implies that A ∩ U ∈ I . So, there exists U ∈ η-O(x) such that A ∩ U ∈ I which is acontradiction.

(iii) Let A,B ⊆ X . Note that A \B ⊆ A and B \A ⊆ B. Then by Theorem 1 (i), (A \B)∗η ⊆ A∗ηand (B \ A)∗η ⊆ B∗η , and so, (A \ B)∗η ∪ (B \ A)∗η ⊆ A∗η ∪ B∗η . Now, by Theorem 1 (iii),
A∗η ∪ B∗η ⊆ (A ∪ B)∗η . Hence, (A \ B)∗η ∪ (B \ A)∗η ⊆ (A ∪ B)∗η .

(iv) Let U ⊆ X . Since U ∩A ⊆ A, by Theorem 1 (i), (U ∩A)∗η ⊆ A∗η , and hence, U ∩ (U ∩A)∗η ⊆
U ∩ A∗η .

(v) Let U ∈ I . Since A ∩ U ⊆ U ∈ I and I is an ideal, A ∩ U ∈ I . Hence, by Theorem 1 (vii),
(A ∩ U)∗η = ∅.

(vi) Let A be an η-closed set. Then A = η-cl(A). Now, note that by Theorem 1 (x), A∗η ⊆
η-cl(A). Hence, A∗η ⊆ A.

(vii) Let A ⊆ X . Since A ∩ A∗η ⊆ A∗η , by Theorem 1 (i), (A ∩ A∗η)∗η ⊆ (A∗η)∗η . Note that byTheorem 1 (vi), (A∗η)∗η ⊆ A∗η . Therefore, (A ∩ A∗η)∗η ⊆ A∗η .
(iix) Let A∪B ∈ I . Since A∪B ∈ I and I is an ideal, A ∈ I and B ∈ I . These imply by Theorem1 (vii), (A ∪ B)∗η = ∅, A∗η = ∅, and B∗η = ∅. Therefore, (A ∪ B)∗η = A∗η ∪ B∗η = ∅.
(ix) Suppose that A∗η 6= η-cl(A∗η). Let η-cl(A∗η) ⊂ A∗η . Then there exists an element x ∈ A∗ηsuch that x /∈ η-cl(A∗η). Note that since x ∈ A∗η , for every U ∈ η-O(x), A ∩ U /∈ I . Now,note that x /∈ η-cl(A∗η). Then x /∈ ⋂{K : K is η-closed and A∗η ⊆ K}. This shows that

x /∈ K for some η-closed set K such that A∗η ⊆ K. This implies that x ∈ Kc for some
η-open set Kc such that Kc ∩ A∗η = ∅. Note that x ∈ Kc and Kc ∩ A∗η = ∅, then itfollows that x /∈ A∗η , and so, for some Kc ∈ η-O(x), A ∩ Kc ∈ I , which is a contradiction.Consequently, A∗η = η-cl(A∗η), then by Remark 1 (iii), A∗η is an η-closed set. Now, notethat A∗η = η-cl(A∗η) and by Theorem 1 (x), hence, A∗η = η-cl(A∗η) ⊆ η-cl(A).

(x) Let A ⊆ A∗η . Suppose that x ∈ η-cl(A). Then x ∈ ⋂{K : K is η-closed and A ⊆ K}.This shows that x ∈ K for every η-closed set K such that A ⊆ K. Note that by Theorem2 (ix), A∗η is an η-closed set. Now, note that since A ⊆ A∗η and A∗η is an η-closed set,

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 6

A∗η ∈ {K : K is η-closed and A ⊆ K}. This implies that x ∈ A∗η . Hence, η-cl(A) ⊆ A∗η . Asa result, by Theorem 1 (x) and 2 (ix), A∗η = η-cl(A∗η) = η-cl(A). �

Theorem 3. Let (X, τ, I) be an ideal topological space where η-O(X) is closed under any two
intersections. Then for any A,B subsets of X , the following properties hold:

(i) (A ∪ B)∗η = A∗η ∪ B∗η;
(ii) for U ∈ η-O(x), U ∩ A∗η = U ∩ (U ∩ A)∗η ⊆ (U ∩ A)∗η; and

(iii) A∗η \ B∗η = (A \ B)∗η \ B∗η ⊆ (A \ B)∗η .

Proof.

(i) Let η-O(X) be closed under any two intersections. Suppose that x /∈ A∗η∪B∗η , then x /∈ A∗ηand x /∈ B∗η implying that there exist U, V ∈ η-O(X) such that A ∩ U ∈ I and B ∩ V ∈ I .Note that A ∩ U ∈ I , B ∩ V ∈ I , and I is an ideal. Then (A ∩ U) ∪ (B ∩ V ) ∈ I . Since
U ∩ V ⊆ U and U ∩ V ⊆ V ,

(A ∩ U) ∪ (B ∩ V ) ⊇
[
A ∩ (U ∩ V )

]
∪
[
B ∩ (U ∩ V )

]
= (A ∪ B) ∩ (U ∩ V ).

It implies that (A ∪ B) ∩ (U ∩ V ) ⊆ (A ∩ U) ∪ (B ∩ V ) ∈ I . Again, since I is an ideal,
(A ∪ B) ∩ (U ∩ V ) ∈ I . Now, note that by assumption, η-O(X) is closed under any twointersections, and so, there exists U ∩ V ∈ η-O(x) such that (A ∪ B) ∩ (U ∩ V ) ∈ I . Thisshows that x /∈ (A ∪B)∗η . Hence, (A ∪B)∗η ⊆ A∗η ∪B∗η . Now, by Theorem 1 (iii), therefore,
(A ∪ B)∗η = A∗η ∪ B∗η .

(ii) Let η-O(X) be closed under any two intersections. For U ∈ η-O(X), suppose that x ∈
U ∩A∗η . Then x ∈ U and x ∈ A∗η . To show that x ∈ (U ∩A)∗η , let V ∈ η-O(x). Since x ∈ Uand U ∈ η-O(X), we can write it as U ∈ η-O(x). Hence, by assumption, U ∩ V ∈ η-O(x).Note that since x ∈ A∗η and U∩V ∈ η-O(x), then A∩(U∩V ) /∈ I for every U∩V ∈ η-O(x).Now, by associativity and commutativity,

A ∩ (U ∩ V ) = (A ∩ U) ∩ V /∈ I

= (U ∩ A) ∩ V /∈ I.

This shows that for every V ∈ η-O(x), (U∩A)∩V /∈ I . It implies that x ∈ (U∩A)∗η . Hence,
U ∩A∗η ⊆ (U ∩A)∗η . Now, note that U ∩A∗η ⊆ (U ∩A)∗η , then U ∩ (U ∩A∗η) ⊆ U ∩ (U ∩A)∗η .Since U ∩ (U ∩ A∗η), by associativity again,

U ∩ (U ∩ A∗η) = (U ∩ U) ∩ A∗η

= U ∩ A∗η.This implies that U ∩ A∗η ⊆ U ∩ (U ∩ A)∗η . In contrast, note that U ∩ A ⊆ A, then byTheorem 1 (i), (U ∩ A)∗η ⊆ A∗η . Thus, U ∩ (U ∩ A)∗η ⊆ U ∩ A∗η . Consequently, as a

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 7result, U ∩ A∗η = U ∩ (U ∩ A)∗η . Note that U ∩ (U ∩ A)∗η ⊆ (U ∩ A)∗η . This shows that
U ∩ A∗η = U ∩ (U ∩ A)∗η ⊆ (U ∩ A)∗η .

(iii) Let A,B ⊆ X . Note that A = (A \ B) ∪ (B ∩ A). Thus A∗η =
[
(A \ B) ∪ (B ∩ A)

]∗
η
. Notethat by assumption, η-O(X) is closed under any two intersections, then by Theorem 3 (i),

A∗η =
[
(A \ B) ∪ (B ∩ A)

]∗
η

= (A \ B)∗η ∪ (B ∩ A)∗η.So, A∗η = (A \ B)∗η ∪ (B ∩ A)∗η . Now, note that A∗η \ B∗η = A∗η ∩
(
B∗η
)c , and since A∗η =

(A \ B)∗η ∪ (B ∩ A)∗η ,
A∗η \ B∗η = A∗η ∩

(
B∗η
)c

=
[
(A \ B)∗η ∪ (B ∩ A)∗η

]
∩
(
B∗η
)c

=
[
(A \ B)∗η ∩

(
B∗η
)c] ∪ [(B ∩ A)∗η ∩

(
B∗η
)c]

=
[
(A \ B)∗η \ B∗η

]
∪
[
(B ∩ A)∗η \ B∗η

]
.

Hence, A∗η \ B∗η =
[
(A \ B)∗η \ B∗η

]
∪
[
(B ∩ A)∗η \ B∗η

]. Note that B ∩ A ⊆ B, then byTheorem 1 (i), (B ∩ A)∗η ⊆ B∗η implies that (B ∩ A)∗η \ B∗η = ∅. Now, since A∗η \ B∗η =[
(A \ B)∗η \ B∗η

]
∪
[
(B ∩ A)∗η \ B∗η

] and (B ∩ A)∗η \ B∗η = ∅, it follows that
A∗η \ B∗η =

[
(A \ B)∗η \ B∗η

]
∪
[
(B ∩ A)∗η \ B∗η

]
=
[
(A \ B)∗η \ B∗η

]
∪∅

= (A \ B)∗η \ B∗η

⊆ (A \ B)∗η.As a result, it shows that A∗η \ B∗η = (A \ B)∗η \ B∗η ⊆ (A \ B)∗η . �

Theorem 4. Let (X, τ, I) be an ideal topological space and A ⊆ X . Then for any η-local function,
the following properties hold:

(i) A∗η ⊆ A∗;
(ii) A∗η ⊆ Γ∗(A); and

(iii) A∗ ⊆ Γ∗(A).

Proof.

(i) Let x ∈ A∗η and U ∈ τ(x). Since every open set is η-open set, U ∈ η-O(x). Also, since
x ∈ A∗η and U ∈ η-O(x), A ∩ U /∈ I . Note that A ∩ U /∈ I and U ∈ τ(x). Hence, A ∩ U /∈ Ifor every U ∈ τ(x), and so, x ∈ A∗. Therefore, A∗η ⊆ A∗.

(ii) Let x ∈ A∗η and U ∈ RO(x). Since every regular-open set is η-open set, U ∈ η-O(x). Also,since x ∈ A∗η and U ∈ η-O(x), A ∩ U /∈ I . Hence, A ∩ U /∈ I for every U ∈ RO(x), and so,
x ∈ Γ∗(A). Therefore, A∗η ⊆ Γ∗(A).

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 8

(iii) Let x ∈ A∗ and U ∈ RO(x). Since every regular-open set is open set, U ∈ τ . Also, since
x ∈ A∗ and U ∈ τ , A ∩ U /∈ I . Hence, A ∩ U /∈ I for every U ∈ RO(x), and so, x ∈ Γ∗(A).Therefore, A∗ ⊆ Γ∗(A). �

Remark 2. Let (X, τ, I) be an ideal topological space and A ⊆ X . Then for any η-local functions,
the following properties hold:

(ii) A∗η ⊆ A∗ ⊆ Γ∗(A);(ii) if η-O(X) = τ , then A∗η = A∗; and(iii) if η-O(X) = RO(X), then A∗η = Γ∗(A)

Theorem 5. Let (X, τ) be a topological space with ideals I1 and I2 on X and A ⊆ X . Then, for
any η-local functions, the following properties hold:

(i) if I1 ⊆ I2, then A∗η
(
I2, η-O(X)

)
⊆ A∗η

(
I1, η-O(X)

)
; and

(ii) A∗η
(

(I1 ∩ I2), η-O(X)
)

= A∗η
(
I1, η-O(X)

)
∪ A∗η

(
I2, η-O(X)

)
.

Proof.

(i) Let I1 ⊆ I2 and x ∈ A∗η(I2, η-O(X)
). Then for every U ∈ η-O(x), A∩U /∈ I2. Since I1 ⊆ I2,

A ∩ U /∈ I1 for every U ∈ η-O(x). Hence, A∗η(I2, η-O(X)
)
⊆ A∗η

(
I1, η-O(X)

).
(ii) Let I1 and I2 be ideals on X . Note that I1 ∩ I2 ⊆ I1 and I1 ∩ I2 ⊆ I2. Then by Theorem 5

(i),
A∗η
(
I1, η-O(X)

)
⊆ A∗η

(
(I1 ∩ I2), η-O(X)

)
and

A∗η
(
I2, η-O(X)

)
⊆ A∗η

(
(I1 ∩ I2), η-O(X)

)
,

and hence,
A∗η
(
I1, η-O(X)

)
∪ A∗η

(
I2, η-O(X)

)
⊆ A∗η

(
(I1 ∩ I2), η-O(X)

)
.

Next, let x ∈ A∗η((I1∩I2), η-O(X)
), then for every U ∈ η-O(x), A∩U /∈ I1∩I2. This impliesthat A∩U /∈ I1 or A∩U /∈ I2. This shows that x ∈ A∗η(I1, η-O(X)

) or x ∈ A∗η(I2, η-O(X)
).Hence, x ∈ A∗η(I1, η-O(X)

)
∪ A∗η

(
I2, η-O(X)

), and so,
A∗η
(

(I1 ∩ I2), η-O(X)
)
⊆ A∗η

(
I1, η-O(X)

)
∪ A∗η

(
I2, η-O(X)

)
.

As a result, thus,
A∗η
(

(I1 ∩ I2), η-O(X)
)

= A∗η
(
I1, η-O(X)

)
∪ A∗η

(
I2, η-O(X)

)
. �

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 94. η-Local Closure

Definition 2. Let (X, τ, I) be an ideal topological space. The η-local closure of A denoted by
Cl∗η(A) is defined by the union of A and the η-local function of A, i.e, Cl∗η(A) = A ∪ A∗η for any
A ⊆ X .

Example 3. Let (X, τ, I) be an ideal topological space where X = {a, b, c}, τ = {∅, X, {a}, {b},
{a, b}}, and I = {∅, {b}}. Then the η-open sets of X are ∅, X , {a}, {b}, {a, b}, {a, c}, and
{b, c}. Let A = {a, b}, Then by Definition 1 and 2, A∗η = {a} and Cl∗η(A) = {a, b} ∪ {a} = {a, b},
respectively.

Theorem 6. Let (X, τ, I) be an ideal topological space and A,B ⊆ X . Then the following properties
hold:

(i) if A ⊆ B, then Cl∗η(A) ⊆ Cl∗η(B);
(ii) Cl∗η(A ∩ B) ⊆ Cl∗η(A) ∩ Cl∗η(B);
(iii) if A is an η-closed set, then Cl∗η(A) = η-cl(A);
(iv) if A ∈ I , then Cl∗η(A) = A;
(v) Cl∗η(A∗η) = A∗η;

(vi) Cl∗η(A) = η-cl(A); and
(vii)

(
Cl∗η(A)

)∗
η

= A∗η .

Proof.

(i) Let A,B ⊆ X and A ⊆ B. By Definition 2, Cl∗η(A) = A ∪ A∗η and Cl∗η(B) = B ∪ B∗η .Since A ⊆ B, by Theorem 1 (i), A∗η ⊆ B∗η . This shows that A ∪ A∗η ⊆ B ∪ B∗η , and hence,
Cl∗η(A) ⊆ Cl∗η(B).

(ii) Let A,B ⊆ X . Since A ∩ B ⊆ A and A ∩ B ⊆ B, by Theorem 6 (i), Cl∗η(A ∩ B) ⊆ Cl∗η(A)and Cl∗η(A ∩ B) ⊆ Cl∗η(B). Hence, it implies Cl∗η(A ∩ B) ⊆ Cl∗η(A) ∩ Cl∗η(B).
(iii) Let A be an η-closed set, then A = η-cl(A). Now, suppose that x /∈ A. It implies that

x /∈ η-cl(A), then x /∈ ⋂{K : K is η-closed and A ⊆ K}. It follows that x /∈ K for some
η-closed set K such that A ⊆ K. Hence, x ∈ Kc for some η-open set Kc such that
A ∩ Kc = ∅. It implies that there exists Kc ∈ η-O(x) such that A ∩ Kc = ∅, and byDefinition of ideal, ∅ ∈ I for any ideal I . Hence, A ∩ Kc ∈ I for some Kc ∈ η-O(x). Thisshows that x /∈ A∗η , and hence, A∗η ⊆ A. It follows that Cl∗η(A) = A ∪ A∗η = A. Note that
A = η-cl(A). Therefore, Cl∗η(A) = η-cl(A).

(iv) Let A ∈ I . Then by Definition 2 and Theorem 1 (vii), Cl∗η(A) = A ∪ A∗η = A ∪ ∅ = A.Consequently, Cl∗η(A) = A.
(v) Let A ⊆ X . Then by Definition 2 and Theorem 1 (vi), Cl∗η(A∗η) = A∗η ∪

(
A∗η
)∗
η

= A∗η. Itfollows that, Cl∗η(A∗η) = A∗η .

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 10

(vi) Let A ⊆ X . Suppose that Cl∗η(A) 6= η-cl(A). Let η-cl(A) ⊂ Cl∗η(A). Then there existsan element x ∈ Cl∗η(A) such that x /∈ η-cl(A). Note that since x ∈ Cl∗η(A), by Definition2, x ∈ A ∪ A∗η implies that x ∈ A or x ∈ A∗η , or both. Suppose x ∈ A∗η . Then for every
U ∈ η-O(x), A ∩ U /∈ I . Now, Since x /∈ η-cl(A), x /∈ ⋂{K : K is η-closed and A ⊆ K}.It follows that x /∈ K for some η-closed set K such that A ⊆ K. Hence, x ∈ Kc for some
η-open set Kc such that A ∩Kc = ∅, and by Definition of an ideal, ∅ ∈ I for any ideal I .It implies that there exists Kc ∈ η-O(x) such that A ∩ Kc ∈ I , and hence, x /∈ A∗η . Also,note that since x ∈ Kc and A ∩ Kc = ∅, x /∈ A. This shows that x /∈ A and x /∈ A∗η , acontradiction.

(vii) Let A ⊆ X . Then by Definition 2 and Theorem 1 (iii), (Cl∗η(A)
)∗
η

=
(
A∪A∗η

)∗
η
⊇ A∗η∪

(
A∗η
)∗
η
.Note that by Theorem 1 (vi), (A∗η)∗η ⊆ A∗η , then A∗η ∪ (A∗η)∗η = A∗η . It implies that A∗η ⊆(

Cl∗η(A)
)∗
η
. Now, let x ∈ (Cl∗η(A)

)∗
η
. Then for every U ∈ η-O(x), Cl∗η(A) ∩ U /∈ I . Now, byDefinition 2, Cl∗η(A)∩U = (A∪A∗η)∩U /∈ I = (A∩U)∪(A∗η∩U) /∈ I . It implies that A∩U /∈ Ior A∗η∩U /∈ I , or both, and so, x ∈ A∗η or x ∈ (A∗η)∗η , or both. It follows that x ∈ A∗η∪(A∗η)∗η .Note that A∗η ∪ (A∗η)∗η = A∗η , and so, x ∈ A∗η . Consequently, (Cl∗η(A)

)∗
η
⊆ A∗η . Thus,(

Cl∗η(A)
)∗
η

= A∗η . �

Theorem 7. Let (X, τ, I) be an ideal topological space and A,B ⊆ X . Then the following properties
hold:

(i) A ⊆ Cl∗η(A) and A∗η ⊆ Cl∗η(A);
(ii) Cl∗η(∅) = ∅ and Cl∗η(X) = X;

(iii) Cl∗η(A) ∪ Cl∗η(B) ⊆ Cl∗η(A ∪ B); and
(iv)

(
Cl∗η(A)

)∗
η
⊆ Cl∗η(A) = Cl∗η

(
Cl∗η(A)

)
.

Proof.
(i) Let A ⊆ X . Note that A ⊆ A ∪ A∗η . Then by Definition 2, A ⊆ Cl∗η(A). Next, note that
A∗η ⊆ A ∪ A∗η , by Definition 2 again, it implies that A∗η ⊆ Cl∗η(A).

(ii) By Definition 2 and Theorem 1 (iv), Cl∗η(∅) = ∅ ∪ (∅)∗η = ∅ ∪∅ = ∅. Next, note that X is auniversal set, then (X)∗η ⊆ X . Hence, by Definition 2, Cl∗η(X) = X ∪ (X)∗η = X .
(iii) Let A,B ⊆ X . By Definition 2 and Thoerem 1 (iii),

Cl∗η(A ∪ B) = (A ∪ B) ∪ (A ∪ B)∗η

⊇ (A ∪ B) ∪ (A∗η ∪ B∗η)

= (A ∪ A∗η) ∪ (B ∪ B∗η)

= Cl∗η(A) ∪ Cl∗η(B).

This shows that Cl∗η(A) ∪ Cl∗η(B) ⊆ Cl∗η(A ∪ B).

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 11

(iv) Let A ⊆ X . Note that by Theorem 6 (vii), (Cl∗η(A)
)∗
η

= A∗η , and by Theorem 7 (i), A∗η ⊆ Cl∗η(A).Hence, it shows that (Cl∗η(A)
)∗
η
⊆ Cl∗η(A). Next, by Definition 2, Cl∗η(Cl∗η(A)

)
= Cl∗η(A) ∪(

Cl∗η(A)
)∗
η
. Note that since (Cl∗η(A)

)∗
η
⊆ Cl∗η(A),

Cl∗η
(
Cl∗η(A)

)
= Cl∗η(A) ∪

(
Cl∗η(A)

)∗
η

= Cl∗η(A).

It follows that, Cl∗η(A) = Cl∗η
(
Cl∗η(A)

). �

Remark 3. The reverse inclusion of Theorem 7 (iii) need not be true in general as shown from the
following example.

Example 4. Let (X, τ, I) be an ideal topological space where X = {a, b, c, d}, τ =
{
∅, X, {c}, {d},

{c, d}
}

, and I =
{
∅, {a}, {b}, {a, b}

}
. Then the η-open sets of X are ∅, X , {c}, {d}, {a, c},

{a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, c, d}, {a, b, d}, and {b, c, d}. Let A = {c} and B = {d}
such that A∪B = {c, d}, Then by Definition 1, A∗η = {c}, B∗η = {d}, and (A∪B)∗η = X . Now, By
Definition 2, Cl∗η(A) = {c}, Cl∗η(B) = {d}, and Cl∗η(A ∪ B) = X . Observe that Cl∗η(A ∪ B) = X

and Cl∗η(A) ∪ Cl∗η(B) = {c, d}. These shows that Cl∗η(A ∪ B) * Cl∗η(A) ∪ Cl∗η(B). Hence, the
above assertion has been verified.

Theorem 8. Let (X, τ, I) be an ideal topological space and A ⊆ X . Then Cl∗η(A) ⊆ Cl∗(A).

Proof. Let A ⊆ X . By Definition 2 and Kuratowski closure operator, Cl∗η(A) = A ∪ A∗η and
Cl∗(A) = A ∪ A∗, respectively. Since by Theorem 4 (i), A∗η ⊆ A∗, A ∪ A∗η ⊆ A ∪ A∗. Hence,
Cl∗η(A) ⊆ Cl∗(A). �

Theorem 9. Let (X, τ, I) be an ideal topological space and A be any subset of X . Then A is an
η-closed set iff A = Cl∗η(A).

Proof. Let A be an η-closed set. Then A = η-cl(A) and by Theroem 6 (vi), Cl∗η(A) = η-cl(A),respectively. Note that A = η-cl(A) and η-cl(A) = Cl∗η(A), then by transitive property, it impliesthat A = Cl∗η(A). Now, on the other hand, let A = Cl∗η(A). Note that by Theorem 6 (vi),
Cl∗η(A) = η-cl(A). Now that A = Cl∗η(A) and Cl∗η(A) = η-cl(A), by transitive property again,
A = η-cl(A). Therefore, A is an η-closed set. �

Note that Theorem 7 (i), (ii), and (iv) satisfy three of the Kuratowski closure axioms. However,Theorem 7 (iii) did not satisfy one of the Kuratowski closure axioms because it is an inclusionproperty. As a result, the following remark is obtained.
Remark 4. The η-local closure, i.e, Cl∗η , need not be a Kuratowski closure operator with respect
to η in general.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.2 12

Theorem 10. Let (X, τ, I) be an ideal topological space where η-O(X) is closed under any two
intersections and A,B ⊆ X . Then Cl∗η(A ∪ B) = Cl∗η(A) ∪ Cl∗η(B).

Proof. Let η-O(X) be closed under any two intersections. Then by Definition 2 and Theorem 3 (i),it follows that
Cl∗η(A ∪ B) = (A ∪ B) ∪ (A ∪ B)∗η

= (A ∪ B) ∪ (A∗η ∪ B∗η)

= (A ∪ A∗η) ∪ (B ∪ B∗η)

= Cl∗η(A) ∪ Cl∗η(B).

Hence, Cl∗η(A ∪ B) = Cl∗η(A) ∪ Cl∗η(B). �

Note that Theorem 7 (i), (ii), (iv) and Theorem 10 using the condition, for any ideal topologicalspaces (X, τ, I) where η-O(X) is closed under any two intersections, satisfy the Kuratowski closureaxioms. As a result, the following remark is obtained
Remark 5. Let (X, τ, I) be an ideal topological space where η-O(X) is closed under any two
intersections, the η-local closure, i.e, Cl∗η , is a Kuratowski closure operator (or Almocera closure
operator) with respect to η.

Note that by Remark 5, Cl∗η is a Kuratowski closure operator (or Almocera closure operator)with respect to η for any ideal topological space (X, τ, I) where η-O(X) is closed under any twointersections. Now, let A be a τ∗η-closed set iff A∗η ⊆ A in any ideal topological space (X, τ, I)where η-O(X) is closed under any two intersections. Then the following lemma is obtained.
Lemma 1. Let A be a τ∗η-closed set iff A∗η ⊆ A in any ideal topological space (X, τ, I) where
η-O(X) is closed under any two intersections. Then A is τ∗η-closed set iff Cl∗η(A) = A.

Proof. Let A be a τ∗η-closed in (X, τ, I) where η-O(X) is closed under any two intersections. Now,since A is a τ∗η-closed, by assumption, A∗η ⊆ A. It follows that A ∪ A∗η = A. Now, by Definiton2, Cl∗η(A) = A ∪ A∗η = A. Therefore, Cl∗η = A. On the other hand, let Cl∗η(A) = A. Now, byDefiniton 2, Cl∗η(A) = A ∪ A∗η = A. So, A ∪ A∗η = A implies A∗η ⊆ A, and so, by assumption, A is
τ∗η-closed. �

Theorem 11. Let (X, τ, I) be an ideal topological space where η-O(X) is closed under any two
intersections. Let τ∗η =

{
J ⊆ X : Cl∗η(Jc) = Jc

}
. Then τ∗η is a topology for X such that τ∗ ⊆ τ∗η

and η-O(X) ⊆ τ∗η .

Proof. Let η-O(X) be closed under any two intersections. Note that by Remark 5, Cl∗η is aKuratowski closure operator with respect to η. Therefore, τ∗η is a topology generated by Cl∗η . Now,to show that τ∗ ⊆ τ∗η , let A be a τ∗-open. Then Ac is a τ∗-closed. Then by Definition of τ∗-closed,

https://doi.org/10.28924/ada/ma.5.2


Eur. J. Math. Anal. 10.28924/ada/ma.5.2 13(
Ac
)∗ ⊆ Ac . Hence, Cl∗(Ac) = Ac ∪

(
Ac
)∗

= Ac implies that Cl∗(Ac) = Ac . Then by Theorem8, Cl∗η(Ac) ⊆ Ac . Now, since Cl∗η(Ac) ⊆ Ac , by Theorem 7 (i), Cl∗η(Ac) = Ac . Hence, by Lemma1, Ac is a τ∗η-closed, and so, A is a τ∗η-open. As a result, thus, τ∗ ⊆ τ∗η . Next, to show that
η-O(X) ⊆ τ∗η , let A be an η-open. Then Ac is an η-closed and by Theorem 9, Ac = Cl∗η(Ac). Itfollows that by Lemma 1, Ac is a τ∗η-closed implies that A is a τ∗η-open. Hence, η-O(X) ⊆ τ∗η . �

5. Conclusion

The concept of the η-local function and the closure Cl∗η has been introduced and demonstratedthrough illustrative examples. Additionally, certain properties have been studied and explored. Itcan be concluded that the closure Cl∗η can only be a Kuratowski closure operator (Almocera closureoperator) if η-O(X) is closed under two intersections. Under this condition, τ∗η can form a topology,making τ∗η a more generalized version of τ∗ and η-O(X).
References

[1] A. Al-Omari, T. Noiri, Local function Γ∗ in ideal topological spaces, Sci. Stud. Res. Ser. Math. Inform. 26 (1) (2016)5-16.[2] E. Hatir, A. Al-Omari, S. Jafari, δ-local functions and its properties in ideal topological spaces, Fasciculi Math. 53(2014) 53-64.[3] D. Jankovic, T.R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly 97 (4) (1990) 295-310.[4] K. Kuratowski, Topology I, Warszawa, 1933.[5] K. Kuratowski, Topology, Academic Press, New York, 1966.[6] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963) 36-41.[7] P.L. Powar, K. Rajak, Some new concepts of continuity in generalized topological space, Int. J. Com. Appl. 38 (5)(2012) 12-17.[8] D. Subbulakshmi, K. Sumathi, K. Indiran, η-open sets in topological, Int. J. Innov. Techno. Explor. Eng. 8 (10S)(2019), 276-282.[9] R. Vaidyanathaswamy, The localization theory in set-topology, Proc. Indian Acad. Sci. Sect. 20 (1944) 51-61.

https://doi.org/10.28924/ada/ma.5.2

	1. Introduction
	2. Preliminaries
	3. -local functions
	4. -Local Closure
	5. Conclusion
	References